Appears in the Proceedings of AAAI-94 Workshop on Case-based Reasoning, 1994
Evaluating the Effectiveness of Derivation Replay in
Partial-order vs State-space Planning
Laurie H. Ihrig & Subbarao Kambhampati
Department of Computer Science and Engineering
Arizona State University, Tempe, AZ 85287-5406
email: laurie.ihrig@asu.edu, rao@asu.edu
Abstract
Case-based planning involves storing individual instances of problem-solving episodes and using them to
tackle new planning problems. This paper is concerned
with derivation replay, which is the main component of a
form of case-based planning called derivational analogy
(DA). Prior to this study, implementations of derivation replay have been based within state-space planning.
We are motivated by the acknowledged superiority of
partial-order (PO) planners in plan generation. Here we
demonstrate that plan-space planning also has an advantage in replay. We will argue that the decoupling of
planning (derivation) order and the execution order of
plan steps, provided by partial-order planners, enables
them to exploit the guidance of previous cases in a more
efficient and straightforward fashion. We validate our
hypothesis through a focused empirical comparison.
Introduction
Recently it has been demonstrated that partial-order planners,
which search in a space of plans, are more flexible and
efficient in plan generation (Barrett and Weld 1994; Minton
et al. 1992). The aim of the current work is to analyze the
possible advantage that a plan-space planner would have in
derivation replay, which is one component of a form of casebased planning called derivational analogy (Carbonell 1986;
Veloso 1992). We are motivated by a desire to see whether the
known advantages of PO planning over state-space planning
in plan generation, also make the former a more efficient
substrate for replay.
Derivational analogy includes all of the following elements
(Carbonell 1986; Veloso 1992): a facility within the baselevel planner to generate a trace of the derivation of a problem
solution, the indexing and storage of the solution trace in a
library of cases, the retrieval of a case in preparation for
solving a new problem, and finally, a replay mechanism by
which the planner can utilize a previous derivation as guidance
This research is supported in part by National Science Foundation under grant IRI-9210997, and ARPA/Rome Laboratory planning initiative under grant F30602-93-C-0039. Thanks to Manuela
Veloso for helpful clarifications regarding Prodigy/Analogy. Thanks
also to Suresh Katukam for his comments. Interested readers are
encouraged to read a companion paper in Proceedings AAAI-94
describing our experiments in greater detail.
to a new search process. The storage and retrieval aspects
of DA remain the same whether we use plan-space or statespace planners. In particular, solutions to the storage problem
such as those proposed in (Veloso 1992) and (Kambhampati
1994) can be used for this purpose. Only the contents of the
trace and the details of the replay component depend on the
underlying planner. Thus, in our evaluation we focus on the
automatic generation and replay of the solution trace.
We will start by comparing plan-space and state-space
planning, two approaches to generative planning which vary
as to how they derive their solutions. Then we will analyze
the relative advantages of doing replay within plan-space vs
state-space planners. One of the difficult decisions faced
by the replay systems is that of deciding when and where
to interleave from-scratch effort with derivation replay (c.f.
(Blumenthal and Porter 1994)). In general, there are no
domain-independent grounds for making this decision. This
makes eager replay, i.e., replaying the entire trace before
returning to from-scratch planning, the most straightforward
strategy. We will show that, for replay systems based on
state-space planners, eager replay inhibits effective transfer
in problems where the plan-step order is critical to the solution.
We will argue that the decoupling of planning (derivation)
order from execution order of steps, provided by plan-space
planners, allows effective transfer to occur with eager replay
in more situations, and thus provides for a more efficient
and straightforward replay framework. We will validate this
hypothesis by comparing plan-space replay to replay systems
implemented on two different state-space planners. We will
end with a discussion of the applicability of our results in the
context of a variety of extensions to the basic replay strategy.
Plan-space vs State-space Planning
State-space planners derive plans by searching through a
space of world states. Each time that a transition is made into
an adjoining state, the operator that facilitates that transition
is added to the end (or, in the case of a backward-search from
the goal state, to the beginning) of the plan that is so far
constructed. This means that steps must be added to the plan
according to their execution order.
As an example, consider the state-space planner TOPI
(Barrett and Weld 1994). TOPI does simple backward search
in a space of states, adding steps to the plan in reverse order of
their execution. A trace of TOPI’s derivation of a solution to a
INITIAL STATE
Goal : (AT-OB OB2 AP1)
Initial : ((IS-A AIRPORT AP1) (IS-A AIRPORT AP2))
(IS-A AIRPORT AP3) (AT-PL PL1 AP3)
(AT-OB OB2 AP2) ...
Name : G1
Name : G4
Type : START-NODE
Type : ESTABLISHMENT
Name : G2
Kind : NEW-STEP
Type : ESTABLISHMENT
New Step: (LOAD-PL OB2 ?P1 ?A2)
Kind : NEW-STEP
Open Cond: ((INSIDE-PL OB2 ?P1) 1)
New Step: (UNLOAD-PL OB2 ?P1 AP1)
Name : G5
Open Cond: ((AT-OB OB2 AP1) GOAL)
Type : ESTABLISHMENT
Name : G3
Kind : NEW-STEP
Type : ESTABLISHMENT
New Step: ((FLY-PL ?P1 ?A3 ?A2)
Kind : NEW-STEP
Open Cond: ( (AT-PL PL1 ?A2) 3))
New Step: (FLY-PL ?P1 ?A2 AP1)
Key to Abbreviations:
Open Cond: ((AT-PL ?P1 AP1) 1)
PL=PLANE,AP=AIRPORT,OB=OBJECT
Final Plan: (FLY-PL PL1 AP3 AP2) Created 4
(LOAD-PL OB2 PL1 AP2) Created 3
(FLY-PL PL1 AP2 AP1) Created 2
(UNLOAD-PL OB2 PL1 AP1) Created 1
Ordering of Steps: ((4 < 3) (3 < 2) (2 < 1))
AP2
AP2
APPLY
AP3
AP3
AP1
AP1
AP2
OP 2
APPLY OP 4
APPLY OP 1
APPLY
AP3
AP2
AP1
AP2
OP 3
AP3
AP1
AP3
AP1
GOAL STATE
Figure 1: An Example Solution Trace for DerTOPI
Figure 2: Moving Through State Space
problem from the logistics transportation domain of (Veloso
1992) is provided in Figure 1. This domain involves the
movement of packages across locations by various transport
devices. The trace corresponds to a simple problem in which
there is an airplane P1 at airport AP3, a package OB2 at airport
AP2, and the goal is to transport this package to destination
AP1. Figure 2 shows graphically how TOPI would derive a
plan for this problem by stepping through a sequence of world
states, applying an operator to the plan at each transition.
Plan-space planners derive their plans by traversing a space
of partly-constructed plans. The plan-space planner moves
through this space by successively modifying the currently
active plan, starting with an empty plan. Plans are refined
by adding constraints which include new steps as well as
new orderings between steps. Figure 4 provides a trace
of SNLP’s decision process in arriving at a solution to our
example problem taken from the logistics domain. SNLP
(McAllester and Rosenblitt 1991; Barrett and Weld 1994) is a
causal-link partial-order planner. A path through plan-space
which corresponds to the derivation contained in Figure 4
is displayed graphically in Figure 3. This figure serves to
illustrate the PO planners’ least commitment strategy when it
comes to step-orderings. Orderings are added as required by
the subgoaling structure, since steps that contribute conditions
must precede the steps that require these conditions. Steporderings are also added to resolve conflicts between steps,
for example, when one step is deleting the contribution of
another. Notice in Figure 3 that this means that new steps may
first be added in parallel to an existing step sequence. Further
step orderings may then be added which accomplish the
interleaving of the new action into the existing plan segment.
It is this ability to interleave new steps into the plan which
gives the PO planner an advantage in replay.
Relative Advantages of PO vs. State-space
Planning in Supporting Replay
The previous section characterizes the differences between
the state-space and plan-space planning formalisms. In this
section, we will look at the ramifications of these differences
on the efficiency of replay.
FINAL
3
PLAN
2
2
1
1
S
S
G
G
4
ADD
2
3
ADD OP ORDER
S
OP 3
1
G
ADD OP 2
ADD
3
OP 4
2
1
1
S
G
4
S
G
ADD OP 1
NULL
PLAN
Figure 3: Moving Through Plan Space
As mentioned earlier, replay involves applying the decisions taken in a previous planning episode to the context of a
new planning problem. Since the new problem may contain
goals and initial conditions which are different from that of
the previous one, it is likely that the result of replay is a plan
which only partially solves the new problem. Thus, the plan
that is returned by the replay procedure may still contain open
conditions, either because a prior decision that was taken in
solving a subgoal is not valid in the new context, or because
there are top level goals that are not covered by the previous
trace. Such partial plans resulting from replay will then have
to be extended into a complete solution for the new problem.
It is in this situation that the plan-space planners show more
flexibility over state-space planners.
In particular, as discussed in the previous section, planspace planners can extend a partial plan by interleaving steps
anywhere in the plan. In contrast state-space planners can
only add steps to either the beginning or the end of the partial
plan. Thus, if the extra goals of the new problem require steps
Goal : (AT-OB OB2 AP1)
Initial : ((IS-A AIRPORT AP1) (IS-A AIRPORT AP2))
(IS-A AIRPORT AP3) (AT-PL PL1 AP3)
(AT-OB OB2 AP2) ...
Name : G1
Name : G7
Type : START-NODE
Type : ESTABLISHMENT
Name : G2
Kind : NEW-LINK
Type : ESTABLISHMENT
New Link: (0 (IS-A AIRPORT AP1) 2)
Kind : NEW-STEP
Open Cond: ((IS-A AIRPORT AP1) 2)
New Step: (UNLOAD-PL OB2 ?P1 AP1)
Name : G8
New Link: (1 (AT-OB OB2 AP1) GOAL)
Type : ESTABLISHMENT
Open Cond: ((AT-OB OB2 AP1) GOAL)
Kind : NEW-STEP
Name : G3
New Step: (LOAD-PL OB2 PL1 ?A4)
Type : ESTABLISHMENT
New Link: (4 (INSIDE-PL OB2 PL1) 1)
Kind : NEW-STEP
Open Cond: ((INSIDE-PL OB2 PL1) 1)
New Step: (FLY-PL ?P1 ?A2 AP1)
Name : G9
New Link: (2 (AT-PL ?P1 AP1) 1)
Type : ESTABLISHMENT
Open Cond: ((AT-PL ?P1 AP1) 1)
Kind : NEW-LINK
Name : G4
New Link: (3 (AT-PL PL1 AP2) 4)
Type : ESTABLISHMENT
Open Cond: ((AT-PL PL1 ?A4) 4)
Kind : NEW-STEP
Name : G10
New Step: (FLY-PL ?P1 ?A3 ?A2)
Type : RESOLUTION
New Link: (3 (AT-PL ?P1 ?A2) 2)
Kind : PROMOTION
Open Cond: ((AT-PL ?P1 ?A2) 2)
Unsafe-link : ((3 (AT-PL PL1 AP2) 4)
Name : G5
2 (AT-PL PL1 AP2))
Type : ESTABLISHMENT
Name : G11
Kind : NEW-LINK
Type : ESTABLISHMENT
New Link: (0 (AT-PL PL1 AP3) 3)
Kind : NEW-LINK
Open Cond: ((AT-PL ?P1 ?A3) 3)
New Link: (0 (AT-OB OB2 AP2) 4)
Name : G6
Open Cond: ((AT-OB OB2 AP2) 4)
Type : ESTABLISHMENT
Key to Abbreviations:
Kind : NEW-LINK
PL = PLANE
New Link: (0 (IS-A AIRPORT AP2) 3)
AP = AIRPORT
Open Cond: ((IS-A AIRPORT ?A2) 3)
OB = OBJECT
Final Plan: (FLY-PL PL1 AP3 AP2) Created 3
(LOAD-PL OB2 PL1 AP2) Created 4
(FLY-PL PL1 AP2 AP1) Created 2
(UNLOAD-PL OB2 PL1 AP1) Created 1
Ordering of Steps: ((4 < 2) (3 < 4) (4 < 1) (3 < 2) (2 < 1))
Figure 4: An Example Solution Trace for DerSNLP
that have to be interleaved into the plan produced through
replay, the PO planner is more likely than the state-space
planner to successfully extend the plan resulting from replay.
As an example, suppose that the trace contained in Figure 4
is replayed for the problem that requires the additional goal,
(AT-OB OB3 AP1), and OB3 is initially on the old route taken
by the airplane. Notice that the optimal way of dealing with
the extra goal requires interleaving two steps -- to load OB3
into the plane and unload it from the plane -- at appropriate
places into the current plan. Replay systems based on a
partial-order planner such as SNLP can accomplish this by
first adding the steps in parallel to the existing steps, and then
ordering them with respect to other steps in the process of
resolving conflicts.
This interleaving of new steps is not possible if replay is
based on state-space planners since they can extend a partial
plan only by adding steps to the beginning or end. In this
particular example, the state-space planners can extend the
partial plan to deal with the additional goal, but do so by
making the plane return to pick up the second package and
then deliver it, thus resulting in a less optimal solution. In
other cases, the inability to interleave the plan may make the
state-space planner backtrack over the partial plan resulting
from replay (This will happen, for example, when solving
problems from the ART-MD-NS domain described below).
To summarize, when replay is based on a state-space
planner and step order is critical for the success of the plan,
eager replay may not allow for effective transfer. Partialorder plan-space planners do not suffer from this problem.
In particular, since the PO planners decouple derivation
(planning) order of plan steps from their execution order, an
eager replay strategy will not mislead them as much. It is
therefore more likely that all of the previous advice that is
applicable to the new problem is fully utilized. This reasoning
leads us to believe that plan-space planners will exhibit greater
performance improvements through replay when interacting
subgoals make step order critical for a plan’s success. The next
section describes an empirical evaluation of this hypothesis.
Empirical Evaluation
An empirical analysis was conducted in order to test our hypothesis regarding the relative effectiveness of eager replay
for PO planners. For a more detailed description of these
experiments, see (Ihrig and Kambhampati 1994). We implemented an eager-replay strategy on the state-space planner,
TOPI (DerTOPI), and the plan-space planner SNLP (DerSNLP). With this strategy, the search process is interrupted
to replay the entire derivation trace before returning to fromscratch planning. We also implemented eager replay on
a second state-space planner, NOLIMIT, since it recently
served as a substrate for a comprehensive derivational analogy system (Veloso 1992). Although NOLIMIT differs from
TOPI in several respects1 , it is also a state-space planner in
the sense that it extends a current partial plan by adding steps
to the end of the plan. To facilitate fair comparisons, the
three planning methods, SNLP, TOPI, and NOLIMIT, were
(re)implemented on the same substrate.
ART-MD-NS Domain: Experiments were run on problems
drawn from two domains. The first was the artificial domain,
ART-MD-NS, originally described in (Barrett and Weld
1994) and shown in the table below:
ART-MD-NS (Dm S 2 ):
A1i precond : Ii add : Pi delete : fIj jj < ig)
A2i precond : Pi add : Gi delete : fIj j8j g [ fPj jj
< ig)
Conjunctive goals from this domain are nonserializable in
that they cannot be achieved without interleaving subplans for
the individual conjuncts. For example, consider the problem
that contains the conjunctive goal G 1 G2 . The subplan for
achieving this goal would be: A 11
A12 A21 A22 . This
plan has to be interleaved with steps to solve the additional
goal G3 . The plan for the new conjunctive goal G1 G2 G3
is A11
A12 A13 A21 A22 A23.
Logistics Transportation Domain: The logistics transportation domain of (Veloso 1992) was adopted for the second set
of experiments. Initial conditions of each problem represented
the location of various transport devices (one airplane and
three trucks) over three cities, each city containing an airport
and a post office. Four packages were randomly distributed
^
! ! !
^ ^
! ! ! ! !
1
NOLIMIT is a means-ends analysis planner like PRODIGY
and STRIPS, attempting goals by backward-chaining from the goal
state. Applicable operators (operators whose preconditions are true
in the current state) are added to the end of the plan and the current
state is advanced appropriately. Unlike STRIPS, NOLIMIT can
defer step addition in favor of further subgoaling. It does this by
adding relevant operators to a list of potential operators before actual
placement in the plan.
Phase
One Goal
%Solved
nodes
time(sec)
Two Goal
% Solved
nodes
time(sec)
Three Goal
% Solved
nodes
time(sec)
Four Goal
% Solved
nodes
time(sec)
ART-MD-NS (depth-first, CPU limit: 100sec)
DerSNLP
DerTOPI
DerNOLIMIT
replay scratch replay scratch
replay scratch
Logistics (best-first, CPU limit: 550sec)
DerSNLP
DerTOPI
replay
scratch
replay
scratch
100%
30
.73
100%
90
.68
100%
30
.45
100%
60
.47
100%
30
.63
100%
120
2.5
100% (3.5)
617
15
100% (3.5)
946
19
100% (5.0)
46
11
100% (3.5)
507
49
100%
257
2
100%
317
2
100%
180
5
100%
184
4
100%
347
8
100%
296
12
100% (5.8)
1571
50
100% (5.8)
2371
51
97% (6.2)
15824
6216
63% (5.6)
8463
3999
100%
395
7
100%
679
4
100%
549
16
100%
462
11
100%
1132
34
100%
662
34
100% (7.9)
6086
230
100% (7.9)
7400
262
0%
-
0%
-
100%
577
35
100%
1204
43
100%
1715
227
100%
1310
96
100%
5324
368
100%
1533
100
100% (10.0)
9864
497
100% (10.0)
24412
1264
0%
-
0%
-
Table 1: Performance statistics in ART-MD-NS and Logistics Transportation Domain (Average solution length is shown in
parentheses next to %Solved for the logistics domain only)
over airports. So as to make step order critical, problems were
chosen from this domain to contain subgoals that interact.
Problems represent the task of getting one or more packages
to a single designated airport.
Whereas each problem in ART-MD-NS has a unique solution, in the logistics domain there are many possible solutions
varying in length. However, optimal (shortest) solutions can
only be found by interleaving plans for individual goals. This
difference has an important ramification on the way eager replay misleads state-space planners in these domains. Specifically, in the ART-MD-NS domain, state-space planners will
have to necessarily backtrack from the path prescribed by eager replay to find a solution. In the logistics domain, they can
sometimes avoid backtracking by continuing in the replayed
path, but will find inoptimal plans in such cases.
Results
The results of testing are shown in Table 1. Each table entry
represents cumulative results obtained from a sequence of 30
problems corresponding to one phase of the run. Problem size
was increased by one goal for each phase. A library of cases
was formed over the entire run. Each time a problem was
attempted, the library was searched for a previous case that
was similar (See (Ihrig and Kambhampati 1994)). If one was
found, the new problem was run both in scratch and replay
mode, and the problem became part of the 30 problem set for
that phase. If there was no previous case that applied, the
problem was merely added to the library.
The first row of Table 1 shows the percentage of problems
correctly solved within the time limit. The average solution length is shown in parentheses for the logistics domain
(Solution length was omitted in ART-MD-NS since all the
problems have unique solutions.) The subsequent rows of
Table 1 contain the total number of search nodes visited for
all of the 30 test problems, and the total CPU time. DerSNLP
was able to solve as many or more of the multi-goal problems
than the two state-space planners both in from-scratch and
replay modes, and did so in less time. Our implementation
of DerNOLIMIT was not able to solve any of the multi-goal
problems in the logistics domain within the time limit, and
this column is therefore omitted from the table.
In the ART-MD-NS domain, replay resulted in performance improvements for DerSNLP which increased with
problem size (See Table 1). Comparative improvements with
replay were not found for the two state-space planners in
the multi-goal phases. In the logistics domain, not only did
DerTOPI fail to improve performance through replay, it also
experienced an increase in average solution length. In contrast, replay in DerSNLP led to performance improvements
(without increasing the solution length). These results are
consistent with our hypothesis that state-space planners will
be misled by eager replay when step order is critical.
Analyzing the Generality of Experimental
Results
The experiments reported in this paper concentrated on the
relative effectiveness of state-space and plan-space planners
in supporting eager replay. Some implemented replay systems, such as REMAID (Blumenthal and Porter 1994) and
PRODIGY/ANALOGY (Veloso 1992) utilize more complex
forms of replay strategies to support derivational analogy.
This raises a question as to the applicability of our experimental results to such frameworks. We will address this issue
below.
Interrupting Replay : Some replay systems interrupt replay
of the trace to interleave from-scratch effort (e.g. (Blumenthal
and Porter 1994)). It would seem as if such strategies could
offset the inflexibility of state-space planners in replaying
cases when step order is critical. Consider again our example
problem from the logistics domain. Recall that we have
a plan that corresponds to transporting a single package to
a designated airport, and the new situation requires us to
transport another package to the same destination, and this
package lies somewhere along the plane’s route. One way
a state-space planner can produce an optimal plan without
backtracking over the result of replay is to interrupt the replay
in the middle, do from-scratch planning to deal with the
additional goal, and resume replay. However, the optimal
point in the derivation for inserting the new steps depends
on the problem description, since it will depend on where
the new package is located on the old route. The early work
on derivation replay that is rooted in state-space planning
has thus been forced to focus a good deal of attention on
the problem of determining when to plan for additional goals
(Blumenthal and Porter 1994). In general, there are no
domain-independent grounds for deciding when and where
the from-scratch problem-solving effort should be interleaved
with replay. This leaves the planner with heuristic guesses,
which may turn out to be incorrect, leading to backtracking
once again. In contrast, as we have shown, the partialorder planner can exploit the previous case with eager replay
without the need to interrupt replay of the trace.
Multi-pass Replay: Other replay systems, such as
PRODIGY/ANALOGY use a multi-pass strategy in replay.
Rather than abandon the case after it has been replayed once,
such systems keep the case and replay it again as necessary when the plan that resulted from replaying that case is
backtracked over. This too could sometimes help state-space
planners in improving their ability to exploit a previous trace.
In our example above, this will involve replaying the old trace
once, and when the planner fails to extend it to deal with the
new goal, backtracking, working on the extra goal, and then
replaying the case again. Veloso (Veloso 1992) discusses
an interesting mechanism for keeping track of the extent to
which the case has been replayed, and moving the pointer up
appropriately whenever backtracking occurs.
Although this is an intuitively appealing idea (and could
in fact be adapted to replay systems based on any planning
framework), it is not guaranteed to completely overcome the
inflexibility of state-space planners in replay. In particular,
unless the planner has domain specific control information,
it may be forced to backtrack many times before finding the
right place to insert the steps corresponding to the new goals.
We thus believe that while the multi-pass replay heuristic may
sometimes improve the effectiveness of replay in state-space
planning, it still does not eliminate the essential advantages
of plan-space planners in supporting reuse.
Multi-case Replay: A feature of PRODIGY/ANALOGY is
its ability to use multiple cases in guiding replay. Although
our empirical study involved only replay of a single case
for each problem, we believe that the results can also be
extended to multi-case replay. When we have multiple cases,
in addition to the decision as to how to interleave replay with
from-scratch planning, we also have the decision as to how
to interleave the replay of the individual cases. When step
order is critical, the state-space planner will have to attempt
both types of interleaving to successfully exploit the guidance
of previous cases. Once again, the plan-space planner will
have less need to do either of these, and can thus not only
replay the cases in any order, but also employ eager replay on
individual cases.
Before concluding this section, we would like to make
some observations about the apparent disparity between our
results and the previous work on DA rooted in state-space
planning which has demonstrated significant performance
improvements with replay (Veloso 1992). We believe the
main reason for this may be the strong presence of interacting
goals in our multi-goal problems, coupled with the fact that we
used vanilla planning algorithms without any sophisticated
backtracking strategies or pruning techniques. Although
these techniques may improve the performance of state-space
planners, these will be of benefit to PO planners as well.
Moreover, we have shown that the shift to plan-space replay
lessens the need for backtracking over the replayed path.
Finally, our results from experiments in the logistics domain indicate that even partial-order planners may be misled
by eager replay in some instances. This happens when the
previous case may have achieved one of the goals using
some step s1 and the new problem contains a goal gn which
cannot be achieved in the presence of s1 . However, in such
cases, state-space planners will also be misdirected. Thus our
hypothesis is only that plan-space planners are less likely to
be misled (and thus more likely to exploit the previous case)
through eager replay.
Summary
In this paper, we described the differences between planspace and state-space approaches to planning. We developed
a testable hypothesis regarding the relative advantage of
plan-space planning over state-space planning in derivation
replay. We supported this hypothesis with the help of a
focused empirical study. We then discussed the generality
and applicability of our hypothesis in the context of a variety
of extensions to the basic replay strategy.
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